186 Chapter 4 Annuities
Example 4.5.3 Assuming that they make all their payments as scheduled, how much
will Pat and Tracy owe on their mortgage after 5 years? 10 years? 15 years? 20 years?
25 years?F ollowing the approach used above, we fi nd after 5 years they will have made 60 payments.
And so 360 60 300 payments remain. The amount they owe is the present value of
those payments. So:PV PMT a _n (^) | (^) i
PV $1072.49 a ___ 300 | (^) .006
PV $1,072.49(138.96827618)
PV $149,042.09
The remaining calculations are essentially the same. The results are displayed in the table
below.
Elapsed Years
Remaining
Payments Annuity Factor
Remaining
Balance
5 300 138.96827618 $149,042.09
10 240 127.00843213 $136,215.27
15 180 109.88446602 $117,849.99
20 120 85.36656977 $91,554.79
25 60 50.26213003 $53,905.63
Note that, as expected, the balance declines much more slowly in the early years than in
later ones. In the first 5 years, it drops from $158,000 down to $149,042, meaning that in
the first 5 years only $9,958 of the original debt is eliminated. In each subsequent 5-year
period, the amount of debt eliminated grows, and in the last 5 years $53,906 is paid off
(since after the final payment the balance must be zero).
Extra Payments and the Remaining Balance
What if Pat and Tracy paid more than their required payments in one or more months?
Suppose that they paid an extra $500 in the first month, but otherwise stuck with the pay-
ment schedule. Can we still use this approach to find out how much they owe later on? The
answer to this question is yes—and no.
It is still true that the amount they owe is the present value of their remaining payments,
and so in theory we could still approach the problem in this way. However, by making that
extra payment, they reduced their balance, and thus shortened the amount of time that it
will take to pay off the loan. After 5 years the number of remaining payments would not
be 300, it would be something smaller. And in that “something” lies the difficulty. It is not
easy to determine the actual number of remaining payments if we deviate from the original
payment schedule. In theory the amount owed is still the present value of the remaining
payments, but in practice, since we cannot readily determine how many payments actually
remain, we will not normally be able to use this approach if there has been a deviation from
the original payment schedule.
Loan Consolidations and Refinancing
In many cases it may be advantageous to replace one or more existing loans with a new
loan. Suppose, for example, that someone started a construction business and borrowed
money to buy needed equipment. Since he was just starting the business, lenders consid-
ered it risky to lend him money, and the best interest rate he could find was 15%. Three
years later, though, his business has been very successful, and he finds that he can borrow
at much better rates. In fact, the very same lender would now make the loan to him at 8%.
Obviously, he would be much happier paying the lower rate.
In most cases, a borrower can take advantage of a more attractive opportunity by
refinancing a loan. Refinancing means paying off an existing debt by taking out a new